Massachusetts Institute of Technology
Department of Physics
8.962 Spring 2006
Problem Set 9
Post date: Thursday, April 27th
Due date: Thursday, May 4th
1. Time scale for variation in microlensing
Consider the microlensing scenario described in the April 20 lecture. (“Microlensing”
is the limit in which the images cannot be separated; one just examines the total
intensity produced by the sum of the two images.) Imagine that the source is a star
in the Large Magellanic Cloud (about 100 kiloparsecs away; 1 pc = 3.26 light years =
3.1 × 1013 km); imagine that the lens is an unseen object of about one solar mass in
the galaxy, roughly 10 kiloparsecs away. If the lens is orbiting in the galactic potential
with a typical galactic orbit speed vorb ∼ 200 km/sec, what is the timescale over which
you expect the light curve to vary? In other words, roughly how long will it take for
the light curve to brighten and then dim?
For students who could not attend that lecture, see Hartle’s textbook, Chapter 11.
2. Proper motion distance
Using the definition of proper motion distance given in Carroll [Eq. (8.124)], compute
dM (z). Your final result should be similar in form to Eq. (8.123) of Carroll and should
confirm the rule that
dL (z) = (1 + z)dM (z) = (1 + z)2 dA (z) ,
where dL (z) is luminosity distance and dA (z) is angular diameter distance.
3. Particle horizons in the early universe
[Note: Carroll Sec. 8.8 provides important background to this and the next two prob­
lems. A very readable and even more comprehensive discussion is given in Chapter 3
of Cosmological Inflation and Large Scale Structure, by Andrew R. Liddle and David
H. Lyth (Cambridge University Press, 2000), and in Chapter 13 of An Introduction to
Modern Cosmology, by Andrew R. Liddle (Wiley, 2003).]
The early universe was so hot and dense that all matter existed in the form of a plasma.
As such, photons did not stream freely, but rather scattered repeatedly off the ionized
charges in the plasma. “Recombination” is the name given to the point at which the
universe cooled sufficiently that electrons and protons could bind into atoms; at this
point, photons can stream freely. The cosmic microwave background (CMB) dates
from the recombination epoch, which is now known to have occurred at a redshift
zr � 1200.
Consider a k = 0 FRW cosmology which is radiation dominated prior to recombination
and is matter dominated afterwards1 . Let rH (tr ) be the maximum coordinate distance
that a photon can travel from the big bang to the epoch of recombination; this is
the “comoving horizon radius” at recombination. (More properly, it is the comoving
particle horizon radius, to be contrasted with the event horizon we will define in about
two weeks.) Only points that are within each other’s particle horizons are in causal
contact.
Let robs (tr ) be the coordinate distance from recombination to an observer on earth;
thus 2 × robs (tr ) is the coordinate separation between patches of the CMB on opposite
parts of the sky.
(a) Compute rH (tr ) and robs (tr ).
The CMB is extremely isotropic; its temperature varies by roughly a part in 105 over
the sky.
(b) Given the isotropy of the CMB, why is the smallness of the ratio rH /robs so puz­
zling? (If you find that this ratio is not small, you might want to revise your answer
to part a!) This is known as the “horizon problem”.
(c) Compute the angular scale θc such that sources separated by θ < θc as seen today
were in causal contact prior to t = tr , but sources separated by θ > θc were not.
Some background: For the remaining two problems, we repeatedly need to refer to
temperature and energy scales commonly used in early universe cosmology. Since
this subject grew out of particle physics, the units of particle physics describe its
relevant quantities — e.g., temperatures are usually given in MeV or GeV. This means
that temperatures are really being reported in the combination kB T , where kB is
Boltzmann’s constant.
The following conversions and definitions will prove useful in problems 4 and 5:
T0 ≡ Mean temperature of universe today (CMB temperature) = 2.73 K
kB ≡ Boltzmann’s constant = 8.6 × 10−11 MeV/K
h̄ = 6.6 × 10−22 MeV sec
π 2 kB4
σ ≡ Stefan-Boltzmann constant =
.
60h̄3 c2
The last constant comes in handy when converting between temperature and energy
density for radiation: A gas of photons2 at temperature T has an energy density
ρ = 4(σ/c)T 4 .
Finally, don’t forget that the temperature of radiation redshifts: Radiation whose
temperature is T today was at temperature a(t)T /a(0) = (1 + z)T at an earlier time t.
1
In fact, the universe became matter dominated somewhat prior to recombination: The redshift at which
matter and radiation contribute equal to the mean density of the universe is zeq � 3000. The error we
make in approximating zeq � zr is small enough for the purposes of this and the next two problems that we
may safely consider the universe to be radiation dominated prior to recombination, and matter dominated
afterwards.
2
If you read the texts by Liddle, you will discover that we are neglecting neutrinos and other relativistic
particles in this analysis. Our conclusions will thus be off by a factor of order unity; fine for our purposes.
4. The horizon problem resolved: The inflationary universe
Suppose that at some very early epoch, t1 ≤ t ≤ t2 (where t1 � t2 � tr ), the universe
resides in a “false vacuum” state: A scalar field φ fills spacetime and provides an
effective vacuum energy ρvac � V (φ). (For more details about how a scalar field of
this type is characterized and evolves, see Carroll pp. 369 – 370.) This vacuum energy
acts like a cosmological constant Λ = 8πρvac . Up to factors of order unity, we may
4
write ρvac � (4σ/c)Tvac
, where kB Tvac ∼ 1015 GeV. (Note: Don’t take this formula for
density as a function of temperature too seriously! The vacuum doesn’t actually have
a temperature, and it is certainly not a gas of photons. This formula is designed to
give this scalar field an energy density that is comparable to the scale of grand unified
theories. We are recycling the photon energy density formula because it gives us a nice
way of understanding the energy scales at which this field operates.)
As the universe expands, the potential slowly evolves. Around time t2 , the scalar
field decays into standard model particles, and the stress energy tensor is no longer
dominated by V (φ). This particles provide the matter and radiation content for our
universe; it is then radiation dominated until recombination, and matter dominated
thereafter.
We will take “slowly evolving” to mean that we can regard V (φ) as a constant from
t1 ≤ t ≤ t2 , and assume that V (φ) � 0 for t > t2 . (See Carroll Sec. 8.8, Liddle and
Lyth Chapter 3, or Liddle Chapter 13 for further discussion.)
(a) By inserting appropriate factors of G and c, and using the fact that 1 MeV =
1.6 × 10−12 gm cm2 /sec2 , compute the value of the effective cosmological constant Λ
in inverse seconds squared.
(b) Repeat the calculation in Problem 3, part (a). Find the value of Ne ≡ Δt(Λ/3)1/2
(where Δt = t2 − t1 ) such that robs (tr )/rH (tr ) = 1. (Ne is often referred to as the
number of “e-foldings” of expansion for reasons that are hopefully obvious.) Estimate
t2 . (You may approximate t1 � 0 for this estimate).
(c) What is the spatial expansion factor, a(t2 )/a(t1 ), during this “inflationary” epoch?
(d) Recalculate the angular scale θc in this inflationary universe.
5. The flatness problem
(a) Use the Friedmann equations to derive a general expression relating values of Ω
at different times — i.e., Ω1 at t1 (corresponding to scale factor a1 ) and Ω2 at t2
(corresponding to a2 ). Do this calculation for a matter dominated universe, and for
a radiation dominated universe. Express your answer in terms of Ω − 1 and the scale
factor. Do not take into account the inflationary physics of Problem 4.
(b) Observations today tell us that the universe is flat with about 1% accuracy; i.e.,
Ω = 1 ± 0.01. Continuing to neglect inflation, estimate the value of Ω − 1 at the epoch
of nucleosynthesis (TN ∼ 1 MeV), and at the presumed epoch of grand unification
(TGUT ∼ 1015 GeV). Given that the universe is flat with at least 1% accuracy today,
how close to flat must it have been in these earlier epochs?
The incredible precision with which the universe must have been flat is called the
“flatness” problem. The early universe’s initial conditions must either have been tuned
with incredible precision to give the universe we see today, or some mechanism must
exist to drive the universe into a nearly flat state.
(c) Repeat part (a) using the inflationary universe of Problem 4.
(d) How does inflation resolve the flatness problem? If the universe passes through
enough “e-foldings” to cure the horizon problem of Problem 4, is it likely to leave the
universe flat enough to account for the observed spatial curvature today?